Consider a grid that has 3 rows of 4 squares in each row with the
lower left corner named A and upper right corner named B. Suppose that
starting at point A you can go one step up or one step to the right at
each move. This is continued until the point B is reached. How many
different paths from A to B are possible?

A permutation f is a 1-1 mapping of the first n positive integers onto
themselves. What is the probability that the permutation has the property
that f(i) = i for at least one value of i, i between 1 and n inclusive?

Find the probability of getting at least 20 duplicate addresses when
drawing a sample of 30,000 at random from UK households (estimate
21,000,000) where there is replacement every time a selection is made.

I have to find a 10 digit number which uses each of the digits 0-9
such that the first digit is divisible by 1, the first two digits make
a number divisible by 2, the first three digits make a number
divisible by 3, and so on up to all ten digits making a number
divisible by 10. I figured it out using mostly guess and check, but
it took a long time. Is there a quicker way?

If there are 8 players and 5 positions, making 6720 different
arrangements possible, how is that number reduced if 2 of the players
can only play on the very outside right, and 1 can only play in the
middle?

A building has 7 elevators, each stopping on at most 6 floors. If you
take the right elevator you can get to any floor from any other floor
without changing elevators. What is the greatest number of floors the
building can have?